Sparse tensor product spectral Galerkin BEM for elliptic problems with random input data on a spheroid
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Sparse tensor product spectral Galerkin BEM for elliptic problems with random input data on a spheroid Alexey Chernov · Duong Pham
Received: 31 May 2013 / Accepted: 13 March 2014 © Springer Science+Business Media New York 2014
Abstract We introduce and analyze a sparse tensor product spectral Galerkin Boundary Element Method based on spherical harmonics for elliptic problems with random input data on a spheroid. Problems of this type appear in geophysical applications, in particular in data acquisition by satellites. Aiming at a deterministic computation of the k-th order statistical moments of the random solution, we establish convergence theorems showing that the sparse tensor product spectral Galerkin discretization is superior to the full tensor product spectral Galerkin discretization in the case of mixed regularity of the data’s k-th order moments, naturally implying mixed regularity of the k-th order moments of the random solution. We prove that analytic regularity of the data’s k-th order moments implies analytic regularity of the solution’s k-th order moments. We illustrate performance of the sparse and full tensor product discretization schemes on several numerical examples. Keywords Sparse spectral discretization · Dirichlet-to-Neumann operator · Random data · Tensor product · Spherical harmonics · Spheroidal coordinates Mathematics Subject Classifications (2010) 65N30 · 35R60 · 41A25
Dedicated to Professor Ernst P. Stephan on the occasion of his 65th anniversary. Communicated by: Ian H. Sloan A. Chernov () · D. Pham Hausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn, Endenicher Allee 64, Bonn 53115, Germany e-mail: [email protected] Present Address: A. Chernov Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK
A. Chernov, D. Pham
1 Introduction In geophysical applications, the Neumann problem exterior to a spheroid where the orbits of satellites are located is of interest; see e.g. Ref. [10] and [12]. Following its orbit, a satellite collects data sets which amount to boundary conditions in scattered points. Obviously, this process is prone to uncertainty for many reasons. One is the error in measurements that cannot be avoided due to imperfect measurement devices. Furthermore, such measurements are usually done by sampling a large but finite number of system snapshots and therefore provide an incomplete information about the system. In this paper we suggest and analyze a natural and efficient numerical approach accounting for uncertainty in the boundary data. As a model problem we consider the Laplace equation with Neumann boundary data in the exterior to a prolate spheroid (the case of an oblate spheroid or a sphere can be treated analogously): ⎧ c ⎪ ⎨U = 0 in D , (1.1) ∂ν U = g on , ⎪ −1 ⎩ as |x| → ∞, U (x) = O |x| where |·| denotes the Euclidean norm and ν denotes the unit outward normal vector on the problate spheroid 2 2 x32 3 x1 + x2 = (x1 , x2 , x3 ) ∈ R : + 2 = 1, a > b >
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